Mathematical nomenclature is a comedy of errors. One mathematician who had the last laugh is John Pell, whose name adorns an equation he never worked on. Unfortunately, this misattribution is all the average mathematician is likely to associate with Pell. In this scholarly tome, Noel Malcolm and Jacqueline Stedall reveal fascinating aspects of this seemingly minor character.
Pell was one of the earliest members of the Royal Society. His collection of mathematical books is second only to that of the Savile Library in the Bodleian. There are many favourable accounts from leading mathematicians of his time: Leibniz wrote of a meeting with ``the famous Mr Pell, a notable mathematician.'' But there is the sense that Pell spent much of his energy on relatively unimportant problems, while some of the most profound developments of 17th-century mathematics passed him by.
What inspired Pell was the notion that the logical, deductive structure of mathematics could serve as a model for the systematic ordering of all knowledge. This project was thwarted by its overly ambitious goals and Pell's reluctance to publish. Nonetheless, it is possible to ``enter into the mental world of this intriguingly awkward and intensely intelligent man.'' Many of his manuscripts have survived, and Pell was an active citizen of the 17th-century `Republic of Letters'. One of his most important correspondents and patrons was Sir Charles Cavendish, brother of the Earl, and later first Duke, of Newcastle. According to John Aubrey's Brief Lives, Cavendish ``was a weake, crooked man, and nature having not adapted him for the Court nor Campe, he betooke himself to the Study of the Mathematiques, wherin he became a great Master.''
Wider circumstances contributed to the extensive nature of Pell's correspondence too. Despairing of the royal cause after the battle of Marston Moor in 1644, Cavendish settled in Hamburg for a number of years. Likewise, Pell peregrinated the Continent for well over a decade, holding professorial appointments in Amsterdam and Breda and serving as Cromwell's special envoy to the Protestant cantons of Switzerland. University administrators should take note of what precipitated Pell's move from the Amsterdam Athenaeum to the rather less illustrious École Illustre at Breda: ``Once they have people here they pay them no attention; were they prepared to work for nothing, or indeed to pay for the privilege, they would be men fit for this city.''
The letters between Pell and Cavendish constitute half of this book and are preceded by an essay on Pell's life and one on his mathematics. The amount of information in these essays and hundreds of footnotes is astounding. Thanks to Pell's catholic interests and the vicissitudes of his life, there is also a wealth of material on the wider history of ideas.
This book is a remarkable scholarly achievement, though the abundance of details and focus on Pell occasionally stand in the way of a good yarn. A fitting example is the controversy between Pell and the Danish mathematician Christian Sørensen Longomontanus over the latter's purported success at circle squaring. The impossibility of constructing with ruler and compass a square equal in area to a given circle was proved only in the late 19th century. Pell's refutation of Longomontanus's claim was based on his formula for the tangent of a double angle. Pell invited the leading mathematicians of his time to supply alternative proofs. One who obliged was the philosopher Thomas Hobbes, later an avid circle-squarer in his own right. Many important sources, notably the letters by Cavendish communicating Hobbes's proof, are collected here. But the non-expert reader may arguably be better served by the account in Douglas Jesseph's Squaring the Circle.
This is not to belittle the authors' feat of providing the first full-length study of Pell. It will undoubtedly become the standard reference for anyone seeking information on Pell's role in the debates of his time. His wide intellectual and geographical engagement makes him a unique witness of 17th-century life.
Pell would have received higher appreciation but for his failure to complete or publish most of his projects. However, as the authors point out, ``intellectual life depends on more than publications; it is made up of innumerable overlapping and interacting presences''. Often we are indebted to Pell unawares, for instance when using the multiplication and division signs he introduced.
Without his prolific correspondence, Pell would have left even less trace. The Italian poet and philosopher Guido Ceronetti wrote a few years ago: ``I passionately sing the praises of letter-writing amongst those thinking beings who have not yet descended to the level of the beasts by communicating solely by telephones and faxes. It is not enough to say homo cogitat. A person who really thinks writes letters to his friends.'' If you are hoping for posthumous recognition, heed his advice.